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Geochimica et Cosmochimica Acta, Vol. 69, No. 3, pp. 593–597, 2005
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doi:10.1016/j.gca.2004.08.005
Calculation of hydrogen isotopic fractionations in biogeochemical systems
ALEX L. SESSIONS1,* and JOHN M. HAYES2
1
Division of Geological and Planetary Sciences, MC 100-23, California Institute of Technology, 1200 E. California Blvd,
Pasadena, CA, 91125 USA
2
Department of Geology and Geophysics, MS #8, Woods Hole Oceanographic Institution, 266 Woods Hole Rd., Woods Hole, MA, 02543 USA
(Received March 10, 2004; accepted in revised form August 5, 2004)
Abstract—Hydrogen-isotopic data are often interpreted using mathematical approximations originally intended for other isotopes. One of the most common, apparent in literature over the last several decades,
assumes that delta values of reactants and products are separated by a constant fractionation factor: ␦p ⫽ ␦r
⫹ ␧p/r. Because of the large fractionations that affect hydrogen isotopes, such approximations can lead to
substantial errors. Here we review and develop general equations for isotopic mass balances that include the
differential fractionation of each component in a mixture and discuss their use in three geochemical
applications. For the fractionation of a single component, the reactant and product are related by ␦p ⫽ ␣p/r␦r
⫹ ␧p/r, where ␣ and ␧ refer to the same fractionation. Regression of ␦p on ␦r should give equivalent
fractionations based on the intercept and slope, but this has not generally been recognized in studies of D/H
fractionation. In a mixture of two components, each of which is fractionated during mixing, there is no unique
solution for the three unknown variables (two fractionation factors and the elemental mixing ratio of the two
hydrogen sources). The flow of H from CH4 and H2O to bacterial lipids in the metabolism of Methylococcus
capsulatus provides an example of such a case. Data and conclusions from an earlier study of that system
(Sessions et al., 2002) are reexamined here. Several constraints on the variables are available based on
plausible ranges for fractionation factors. A possible refinement to current experimental procedures is the
measurement of three different isotopes, which would allow unique determination of all variables. Copyright
© 2005 Elsevier Ltd
1983; Criss, 1999). The subscript T designates the total derived
from mixing components 1, 2...n. For the specific case of
hydrogen isotopes, X refers to total hydrogen (D ⫹ H) and F
refers to the fractional abundance of deuterium [⫽ D/(D ⫹ H)].
When mixing is accompanied by isotopic fractionation, application of Eqn. 1 is not straightforward. A concrete example
is a cell that is synthesizing lipids while using two isotopically
distinct sources of hydrogen, e.g., methane and water. Isotopic
fractionations will accompany the transfer of H from each of
those sources to lipids. Mixing calculations become complicated because there is no form of the fractionation factor (␣)
that can be used directly to transform values of F. This difficulty leads to a more useful, approximate form of the massbalance equation
1. INTRODUCTION
Recent improvements in analytical methodology have produced a rising tide of interest in hydrogen-isotopic studies. In
dealing with the results, most authors have used mathematical
approximations originally adopted for 13C and 18O, for example that delta values of two components related by a single
fractionation will have a 1:1 relationship: ␦a ⫽ ␦b ⫹ ␧a/b,
or ␦a ⫽ ␦b ⫹ 1000ln␣a/b. Because of the very large fractionations that affect hydrogen isotopes, particularly in many biologic systems, such approximations are inappropriate. In at
least one case (Sessions et al., 2002), interpretive errors have
resulted. In many others, confusion has or will result from the
publication of equations in which approximations are overlooked. Accordingly, we review here the mathematics of isotopic fractionation with specific reference to hydrogen, examining several specific biogeochemical applications. The
equations derived and conclusions reached are applicable to all
isotopic systems.
RT ⫽ X1R1 ⫹ X2R2 ⫹ ... XnRn
where R is the isotope ratio (e.g., D/H). Eqn. 2 is exact in the
limit that R (and hence F) approaches zero. Errors increase in
proportion to both the absolute isotope ratio (R) and to the
differences between isotope ratios. Thus relatively large isotope
ratios can be tolerated as long as fractionations are small, as is
the case for 13C where R is ⬃0.01 and the range of R values is
generally ⬍0.0005 (⌬R/R ⬍ 50‰). Conversely, large fractionations can be tolerated as long as isotopic abundances are low,
as for 2H where R ⫽ 0.00015 and the range of R values can be
0.00008 (⌬R/R ⫽ 500‰) or larger. An alternative form of
Eqn. 2, in which values of X represent the mole fractions of the
reference isotope, e.g. 12C or 1H, is exact for all cases and has
been suggested by Criss (1999).
Eqn. 2 can be modified to accommodate fractionations of
individual components. If Rn is the isotope ratio of the source
2. GENERAL EQUATIONS
Isotopic mass balance in a mixture of components can be
described without approximation, regardless of the levels of
isotopic enrichment, by
FT ⫽ X1F1 ⫹ X2F2 ⫹ ... XnFn
(1)
where X is the molar fraction of each component of the mixture,
and F is the fractional abundance of the rare isotope (Hayes,
* Author to whom
([email protected]).
correspondence
should
be
(2)
addressed
593
594
A. L. Sessions and J. M. Hayes
Table 1. Calibrations between ␦D values of lipids and environmental water.
Analyte
Derived relationship
␣slope(a)
␣intept(b)
Bulk lipids
Phytoplantonic sterols
Palmitic acid
␦1 ⫽ 0.546␦w ⫺ 141‰
␦1 ⫽ 0.748␦w ⫺ 199‰
␦1 ⫽ 0.939␦w ⫺ 167‰
0.546
0.748
0.939
0.859
0.801
0.833
a
b
R2
0.618
0.965
0.894
Reference
Sternberg (1988)
Sauer et al. (2001)
Huang et al. (2002)
Fractionation factor calculated from the slope of the derived relationship.
Fractionation factor calculated from the intercept of the derived relationship.
of the nth component, then the isotope ratio of that component
in the mixture will be ␣nRn, where ␣n is the fractionation factor.
Eqn. 2 then becomes
RT ⫽ X1␣1R1 ⫹ X2␣2R2 ⫹ ... Xn␣nRn
(3)
A second common modification of Eqn. 2 substitutes delta
values (␦) for isotope ratios (R). Here we follow the lead of
Farquhar et al. (1989) and Mook (2000) who point out that the
ubiquitous permil symbol implies a factor of 103, which can
then be omitted from the definition of delta: ␦x ⫽ Rx/Rstd ⫺ 1.
This simplification removes factors of 1000 that would otherwise clutter the equations presented below. Substituting in Eqn.
3 gives
共␦T ⫹ 1兲Rstd ⫽ X1␣1共␦1 ⫹ 1兲Rstd ⫹ ... Xn␣n共␦n ⫹ 1兲Rstd
(4)
which simplifies to
␦T ⫹ 1 ⫽ X1共␣1␦1 ⫹ ␣1兲 ⫹ ... Xn共␣n␦n ⫹ ␣n兲
(5)
Using the identity X1 ⫹ X2 ⫹. . .Xn ⫽ 1 then gives
␦T ⫽ X1共␣1␦1 ⫹ ␣1 ⫺ 1兲 ⫹ ... Xn共␣n␦n ⫹ ␣n ⫺ 1兲
(6)
A third common substitution is to use ␧ in place of the fractionation factor ␣, where ␧ ⫽ ␣ ⫺ 1 (again assuming that
permil units imply a factor of 103). Eqn. 6 thus becomes
␦T ⫽ X1共␣1␦1 ⫹ ␧1兲 ⫹ ... Xn共␣n␦n ⫹ ␧n兲
(7)
where ␣n and ␧n represent the same fractionation.
Despite the use of the delta notation, Eqn. 7 does not involve
any approximation beyond that inherent in substituting isotope
ratios (Eqn. 2) for fractional abundances. Thus for systems
containing a natural abundance of D, 13C, 15N, or 18O, Eqn. 7
can be viewed as exact and should be used whenever possible,
remembering that values of ␦ and ␧ in permil units must be
divided by 1000 before insertion into Eqn. 7.
3. SOME COMMON GEOCHEMICAL APPLICATIONS
ically be as large as 18 (Bigeleisen and Wolfsberg, 1958).
If ␦P is plotted as a function of ␦S, the magnitude of the
fractionation can be determined from the intercept (␧), the
slope (␣), or both. In fact, their values should be redundant,
with ␧ ⫽ ␣ ⫺ 1. As a concrete example of this phenomenon,
we consider the fractionation of hydrogen isotopes by plants
during photosynthesis. Because water is the source of all
hydrogen in plants, the isotopic compositions of plant lipids
and environmental water are expected to be related simply
by Eqn. 8, in which a single fractionation factor represents
the net effect of all biosynthetic processes. Several authors
have attempted to calibrate this fractionation by measuring
values of ␦D for plant lipids from lakes covering a latitudinal range. Results are summarized in Table 1. In each case
the intercept of the regression has been interpreted as the
‘true’ fractionation factor. However, there is significant disagreement between the fractionations implied by the slope
(␣) and intercept (a ⫺ 1) of each of the regressions. The
fractionations should be equivalent, and the fact that they are
not implies that the relationship is not as simple as has been
assumed. Agreement between the slope and intercept improves as the correlation coefficient for each regression
increases, suggesting that the estimate of Sauer et al. (2001)
is the closest to representing a single process. As with any
statistically significant correlation, the derived equations can
still be used to predict values of ␦D for environmental water
based on those of sedimentary lipids, but they should not be
interpreted as representing a single fractionation between
water and lipid.
3.2. Fractionation of One Component in a
Two-Component Mixture
If two components are being mixed but isotopic fractionation
affects only one of them, Eqn. 7 can be reduced to
␦ T ⫽ X 1共 ␣ 1␦ 1 ⫹ ␧ 1兲 ⫹ 共 1 ⫺ X 1兲 ␦ 2
(9)
3.1. Fractionation of a Single Component
For the fractionation of a single component Eqn. 7 simplifies
to
␦ P ⫽ ␣␦S ⫹ ␧
(8)
where ␦P and ␦S represent the product and source, respectively.
As ␣ approaches unity, the frequently-used approximation
␦P ⫽ ␦S ⫹ ␧ becomes accurate. This approximation serves for
carbon and oxygen isotopes, where values of ␣ are typically
between 0.95 and 1.05, but is inappropriate for hydrogen where
values of ␣ commonly range from 0.7 to 1.5 and can theoret-
A common application of Eqn. 9 arises in the analysis of
organic hydrogen in samples containing exchangeable H
(Schimmelmann et al., 1999; Wassenaar and Hobson, 2000).
After equilibrating such a sample with water of known D/H
ratio, the exchangeable hydrogen will have a new isotopic
composition offset from that of the water by some fractionation
factor, while C-bound H will retain its original isotopic composition. Using the notation of Schimmelmann (1991) and
Wassenaar and Hobson (2000), Eqn. 9 can be rewritten as
␦T ⫽ f e共␣e/w␦w ⫹ ␧e/w兲 ⫹ 共1 ⫺ f e兲␦n
(10)
Calculation of D/H fractionations
Table 2. Hypothetical values for parameters in Eqn. 11 showing that
it does not have a unique solution.
Experimental dataa
␦T (‰)
␦1 (‰)
␦2 (‰)
⫺240
⫺280
⫺320
0
⫺100
⫺200
⫺200
⫺200
⫺200
Possible solutionsb
X1
␣1
␣2
0.40
0.45
0.50
0.60
etc.
1.000
0.889
0.800
0.667
0.750
0.818
0.900
1.125
595
unknown coefficients X1, ␣1, ␣2. All are exact solutions for the
three equations defined by the top half of the table. They are
merely examples, and there is an infinite number of such
solutions. Accordingly, the approach taken by Sessions et al.
(2002) to provide a unique solution must be invalid. To redress
that fault, we first consider alternative approaches, then, in
section 3.5 below, reconsider the conclusions reached by Sessions et al. (2002).
The set of (X1, ␣1, ␣2) solutions for Eqns. 11 and 12 has a
single degree of freedom, i.e., fixing the value of any one of the
three parameters uniquely defines the other two. If none of the
three parameters can be estimated independently, the simplest
approach to this problem lies in reducing the number of free
variables by reparameterizing Eqn. 12 as
R T ⫽ p 1R 1 ⫹ p 2R 2
a
Hypothetical data chosen to follow the form of Eqn. 11.
Possible values for the variables in Eqn. 11, all of which satisfy the
experimental data listed above.
b
where fe is the fraction of exchangeable hydrogen, and the
subscripts e, n, and w refer to exchangeable, nonexchangeable,
and water H, respectively.
Several recent reports have dealt with methodologies for
determining both fe and ␦n. Unfortunately, exact formulations
(eqns. 1– 4 in Schimmelmann et al., 1999; eqns. 1–3 in
Wassenaar and Hobson, 2000) and approximate formulations
(eqns. 1 and 2 in Schimmelmann, 1991; eqns. 2–3 in Chamberlain et al., 1997; eqn. 1 in Hobson et al., 1999; eqn. 4 in
Wassenaar and Hobson, 2000) have both been reported, leading
to the potential for considerable confusion. In all cases, calculations based on Eqn. 10 (above) or eqn. 4 in Schimmelmann et
al. (1999) will give the most accurate results.
3.3. Fractionation and Mixing of Two Components
A more general expression reflecting the potential fractionation of both components is
␦T ⫽ X1共␣1␦1 ⫹ ␧1兲 ⫹ 共1 ⫺ X1兲共␣2␦2 ⫹ ␧2兲
(11)
The equation can be written using isotope ratios rather than
delta values as
R T ⫽ X 1␣ 1R 1 ⫹ 共 1 ⫺ X 1兲 ␣ 2R 2
(12)
Eqns. 11 and 12 are equivalent, and provide identical accuracy.
They describe a situation that is encountered, for example,
when studying hydrogen isotopic fractionations in heterotrophic organisms, all of which must have at least two potential
sources of hydrogen, i.e., organic substrates and water (Hobson
et al., 1999; Sessions et al., 2002; Valentine et al., 2004).
Eqn. 12 contains terms in which X and ␣ are multiplied, and
so is nonlinear. As a result, multiple solutions for (X1, ␣1, ␣2)
exist. This is true no matter how many parallel experiments are
conducted with differing values for R1 and R2. This fact is
demonstrated in Table 2, which provides three sets of hypothetical isotopic compositions. All conform to Eqn. 11 and thus
represent data that might be collected in an experiment designed to provide three equations with only three unknowns.
The bottom half of Table 2 lists four sets of values for the
(13)
where p1 ⫽ X1␣1 and p2 ⫽ (1 ⫺ X1)␣2. Eqn. 13 is linear and
has a unique solution, thus both parameters (p1, p2) can be
determined directly by the regression of RT on R1 or R2.
Although values for the original variables (X1, ␣1, ␣2) still
cannot be uniquely determined, several useful constraints can
be recognized based on the properties of the physical quantities
being studied. These include
1. The value of X1 must lie between zero and one. Thus p1
provides a minimum estimate for ␣1, and p2 provides a minimum estimate for ␣2.
2. Both fractionation factors must have values ⬎0. Thus if
either p1 or p2 ⫽ 0, it can be concluded that X1 ⫽ 0 or X1 ⫽ 1,
respectively.
3. Fractionation factors for most natural systems vary over
limited ranges, even for hydrogen isotopes. For example, fractionation between water and organic hydrogen generally ranges
between 0.900 and 0.650 (Estep and Hoering, 1980; Sessions et
al., 1999). These ranges can be used to further constrain possible values of X1. The constraints placed on X1 by assumed
ranges for ␣1 and ␣2 should be overlapping but not identical.
The form of Eqn. 13 leads to two additional considerations
regarding experimental design. First, both parameters (p1, p2)
can be calculated if the isotopic composition of only one
hydrogen source is varied, and that of both hydrogen sources is
known. This provides significant analytical convenience, since
it is easy to change the isotopic composition of water by the
addition of D2O, but difficult to change the isotopic composition of an organic substrate.
Second, the slope of any regression can be calculated with
smaller uncertainty than the intercept, except in the particular
case where the intercept value lies near the centroid of the
independent variable data. Thus for experiments with only one
hydrogen source (section 3.1 above), it is generally more accurate—and never less accurate—to estimate a fractionation
factor from the slope of the regression than from the intercept.
3.4. Experiments Using Three Isotopes
In principle, the problem can be solved by measuring abundances of three or more isotopes, e.g., protium, deuterium, and
tritium. Isotope effects for most chemical reactions vary with
isotopic mass following the relationship
3
␣ ⫽ 共 2␣ 兲
m
(14)
596
A. L. Sessions and J. M. Hayes
Table 3. Values of the coefficients p1 and p2 for lipids from
Methylococcus capsulatus, reproduced from Sessions et al. (2002).
Lipid
p1a
p2b
squalene
4-methyl sterol ⫹ diplopterol
4,4-dimethyl sterol
hopanol
3-methyl hopanol
16:1 fatty acid (AS)c
16:1 fatty acid (PL)c
16:0 fatty acid (AS)
16:0 fatty acid (PL)
0.220
0.247
0.253
0.238
0.280
0.305
0.278
0.322
0.327
0.601
0.578
0.582
0.583
0.578
0.626
0.653
0.650
0.646
Equivalent to fm␣l/m in table 4 of Sessions et al. (2002).
b
Equivalent to (1 ⫺ fm)␣l/w.
c
AS ⫽ acetone soluble fraction (neutral lipids); PL ⫽ phospholipid
fraction.
a
The left superscripts 2 and 3 indicate fractionation factors
affecting the D/H and T/H ratios, respectively. The constant m
is related only to the masses of the isotopes involved (Bigeleisen, 1965), and for the hydrogen system has a theoretical
value of 1.442 (Swain et al., 1958; Cleland, 2003). In contrast,
the variable X is a characteristic of each hydrogen source, and
its value does not vary between isotopes. Thus for an experiment in which both D/H and T/H ratios are measured, we can
write two mass balance equations
RT ⫽ X1( 2␣1)(2R1) ⫹ (1 ⫺ X1)(2␣2)(2R2)
(15a)
RT ⫽ X1( 2␣1)m(3R1) ⫹ (1 ⫺ X1)(2␣2)m(3R2)
(15b)
2
3
version of Eqn. 11 in which values of ␣ were set to 1.0 while
values of ␧ were allowed to vary, Sessions et al. (2002) calculated values for X1 (termed fm). That approximation is invalid
and the results must be discarded. Here we reinterpret the data
in Table 3 in light of the equations developed above.
Given values of p1 and p2, X1 can be determined only if
either ␣1 or ␣2 is known:
X1 ⫽ p1 ⁄ ␣1
(19)
X 1 ⫽ 共 ␣ 2 ⫺ p 2兲 ⁄ ␣ 2
(20)
and
Then, given X1, the remaining ␣ value can be calculated. If
neither ␣ value is known precisely, as is the case here, combination of Eqns. 19 and 20 yields the relationship which must
prevail between the fractionation factors:
␣ 1 ⫽ p 1␣ 2 ⁄ 共 ␣ 2 ⫺ p 2兲
(21)
Two families of curves result and are shown in Figure 1. One
represents the relationship that must prevail between ␣1 and ␣2
for isoprenoidal products, the other for fatty acids. For any
given (␣1, ␣2), there is a corresponding X1 which is also shown
in Figure 1.
The measured isotopic compositions require that ␣1 and ␣2
are related inversely. If the lipid-water fractionations are similar to those observed in plants (Sessions et al., 1999), then
which can be parameterized as
2
R T⫽ 2 p 1· 2R 1⫹ 2 p 2· 2R 2
(16a)
3
R T⫽ 3 p 1· 3R 1⫹ 3 p 2· 3R 2
(16b)
The p coefficients can be determined by linear regression, and
algebraic combination of the two equations then gives
共2␣1兲m⫺1 ⫽ 3p1 ⁄ 2p1
(17)
共2␣2兲m⫺1 ⫽ 3p2 ⁄ 2p2
(18)
and
Unfortunately, sample requirements for precise measurement
of T/H ratios are thousands of times larger than those required
for measurement of D/H ratios. Still, the approach does offer
one potential route to determining hydrogen-isotopic fractionation factors in organisms that obligately use two or more
hydrogen sources.
3.5. Interpretation of Data from Methylococcus capsulatus
In an earlier report regarding the methanotrophic bacterium
Methylococcus capsulatus, Sessions et al. (2002) systematically varied ␦1 and ␦2 (corresponding to ␦CH4 and ␦H2O) and
measured values of ␦T (corresponding to ␦lipid) in multiple,
parallel cultures. To interpret the results, values for p1 and p2
(termed a and b in Sessions et al., 2002) were calculated by
multivariate regression. This part of the analysis is correct, and
the data are reproduced in Table 3. Next, starting from a
Fig. 1. Relationships between ␣1 and ␣2, equivalent to ␣lipid/methane
and ␣lipid/water, respectively. The lines derive from Eqn. 21 and the
values of p1 and p2 specified in Table 3. The solid lines pertain to the
polyisoprenoidal compounds (squalene, sterols, hopanols). The dotted
lines pertain to the fatty acids. The heavy, vertical broken lines mark
the ranges of ␣2 observed in plants (Sessions et al., 1999). Different
ranges are shown because the polyisoprenoidal compounds are generally depleted in D (0.690 ⱕ ␣2 ⱕ 0.792) relative to the fatty acids
(0.774 ⱕ ␣2 ⱕ 0.830). The italicized numbers and tick marks normal
to the fractionation lines mark values of X1.
Calculation of D/H fractionations
values of ␣1, the lipid-methane fractionation factor, are probably near 1.3. This would indicate that, when H was removed
from methane, protium flowed preferentially to fates other than
lipid biosynthesis, leaving deuterium to be concentrated in the
lipids. In these circumstances, X1 ⬇ 0.2, i.e., only ⬃20% of
lipid hydrogen is derived from CH4. Conversely, if the lipidmethane fractionation approached 0.570, the intermolecular
isotope effect associated with methane uptake by the methane
monooxygenase enzyme in vitro ( Wilkins et al., 1994), fractionations between lipids and water must have been ⬎1.0.
Values of X1 would be near 0.5.
The first of these alternatives is more likely. It is based on
observed water-to-lipid fractionations. The fractionation used
to provide a boundary condition in the second alternative
pertains to methane uptake rather than to biosynthesis, and
takes no account of fractionations downstream from methane
oxidation. If the former interpretation is correct, then the data
are consistent with the earlier conclusion (Sessions et al., 2002)
that X1 remains approximately constant for all lipids, during
both exponential and stationary phases of growth.
4. CONCLUSIONS
Mathematical approximations originally developed for use in
manipulating carbon- and oxygen-isotopic data can introduce
large errors when used for D/H data. In systems with only one
hydrogen source, the reactant and product delta values must be
linearly related, and the slope and intercept of that relation
should indicate equivalent fractionations. In systems with two
or more hydrogen sources, in which both sources have been
fractionated from their original isotopic composition, it is not
possible to uniquely determine both fractionation factors and
the relative contributions of each hydrogen source to the mixture from measurements of D/H ratios alone. This is true even
when the isotopic compositions of all hydrogen sources are
varied independently. Experiments in which the relative abundances of three isotopes are measured have the ability to solve
this problem.
Acknowledgments—The authors gratefully acknowledge the helpful
advice and comments of Tapio Schneider. Associate editor Jeffrey
Seewald, reviewer David Valentine, and two anonymous reviewers
provided many helpful suggestions. A.L.S. is supported by NSF EAR0311824, and J.M.H. is supported by the NASA Astrobiology Institute.
Associate editor: J. Seewald
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